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Quaternions
"In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 184312 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space3 or equivalently as the quotient of two vectors.4 Quaternions are generally represented in the form: : a'' + ''b'i'' + c'j'' + d'''k' where ''a, b'', ''c, and d'' are real numbers, and 'i', 'j', and 'k''' are the fundamental quaternion units." - "My journey through quaternions, some observations, some questions. So if you like maths then read it, get involved, help. It is fascinating, it is profound!" "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." (William Rowan Hamilton) ‎"If quaternions are used consistently in theoretical physics, we get a complete description of the physical world, with relativistic and quantum effects easily taken into account. Thus Hamilton’s idea, which motivated more then half of his professional life, that quaternions are a fundamental building block of the physical universe, appears to be essentially correct in the light of contemporary knowledge."" Octonions :"In 1843 Sir William Rowan Hamilton invented the quaternions, carving their defining equations into the stones of Brougham Bridge in Dublin Ireland. Simultaneously the cross product and dot product were born, in one unified package." :"Within one year of publication, the quaternion notation was generalized to octonions. This algebra is a non-associative normed division algebra, and a Moufang loop (similar to associativity, but weaker). Thus the properties of nature requiring of bra-ket notation are naturally embedded in the octonions. :Further, the split-complex numbers and split-quaternions (historically also called coquaternions) were known by 1849. Had the quaternion notation won the day, these algebras would have been familiar when the Lorentz transformations were derived. In one dimension, the algebra of the Lorentz transformations are identical to to the split-complex numbers." Hyper-complex (Quaternion) indices (see also Optics#Hyper-complex (Quaternion) indices) |arXiv:/Adler2017/Does the Peres experiment using photons test for hyper-complex (quaternionic) quantum theories?> "Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the S-matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the S-matrix, the forward scattering amplitude for electromagnetic wave scattering, and the index of refraction, we show that the index of refraction is necessarily complex, not quaternionic. This implies that the recent optical experiment of Procopio et al. based on the Peres proposal does not test for hyper-complex or quaternionic quantum effects arising within the standard Hilbert space framework. Such a test requires looking at near zone fields, not radiation zone fields." |BenPrather:/2012/Split-Octonion Physics> "In 1843 Sir William Rowan Hamilton invented the quaternions, carving their defining equations into the stones of Brougham Bridge in Dublin Ireland. Simultaneously the cross product and dot product were born, in one unified package. By the 1880 Josiah Gibbs and Oliver Heaviside had developed a competing vector notation, with the added benefit that the dot product could be generalized to arbitrary dimensions. This utility won the day, and lead to tensor arithmetic that biases the mathematical language in favor of higher order dimensions and the possibility of curved metrics. The deal was sealed with general relativity’s use of curved space-time to derive the precession of Mercury. Unfortunately, this also leads to difficulties. We must eventually learn to distinguish between polar and axial vectors, leading to the notion of tensors and pseudo-tensors. At the quantum mechanical level this leads to non-associative behavior that can not be expressed in the normally associative tensor arithmetic." "Further, the split-complex numbers and split-quaternions (historically also called coquaternions) were known by 1849. Had the quaternion notation won the day, these algebras would have been familiar when the Lorentz transformations were derived. In one dimension, the algebra of the Lorentz transformations are identical to to the split-complex numbers. Adding additional split-roots to this algebra, similar to how the octonions were constructed, leads to the split-octonions. This algebra represents a scalar, an axial vector, a polar vector and a pseudo-scalar in one algebraic format. Aside for its zero divisors, with norm 0, this algebra has many of the same algebraic properties of the octonions. This algebra, however, is Lorentzian. In fact, the zero divisors can be seen to have the same singular properties as light cones." |ResearchGate:/Weng2018/Varying speed of light and field potential in the octonion spaces> "The paper focuses on exploring the influence of the electromagnetic scalar potential on the speed of light in the octonion spaces. J. C. Maxwell was the first to introduce the quaternions to research the physical quantities of electromagnetic fields. The subsequent scholars apply the quaternions and octonions to grope for the physical quantities of electromagnetic and gravitational fields, including the transformation between two coordinate systems. In the octonion space, the radius vector can be combined with the integrating function of field potential to become one composite radius vector. The latter is considered as the radius vector in an octonion composite-space, which belongs to the function spaces. In the octonion composite-space, when there is the relative motion between two coordinate systems, it is capable of deducing the Galilean-like transformation and Lorentz-like transformation. From the two transformations, it is able to achieve not only the influence of the relative speed on the speed of light (or Sagnac effect), but also the impact of the electromagnetic scalar potential on the speed of light. The study reveals that the electromagnetic scalar potential has a direct influence on the speed of light in the optical waveguides. Up to now, the contribution of the electromagnetic scalar potential on the speed of light has never been inspected in the experiments. The paper appeals intensely to validate this inference in a few relevant experiments, revealing further some new physical quantities of refractive indices in the optical waveguides." Category:Mathematics Category:Number Theory